LNCS 7042 - The Dissimilarity Representation for ... · The Dissimilarity Representation for...

The Dissimilarity Representation

for Structural Pattern Recognition

Robert P.W. Duin1 and Elzbieta P ↪ekalska2

1 Pattern Recognition Laboratory,Delft University of Technology, The Netherlands

[email protected] School of Computer Science,

University of Manchester, United [email protected]

Abstract. The patterns in collections of real world objects are often notbased on a limited set of isolated properties such as features. Instead, thetotality of their appearance constitutes the basis of the human recog-nition of patterns. Structural pattern recognition aims to find explicitprocedures that mimic the learning and classification made by humanexperts in well-defined and restricted areas of application. This is oftendone by defining dissimilarity measures between objects and measuringthem between training examples and new objects to be recognized.

The dissimilarity representation offers the possibility to apply thetools developed in machine learning and statistical pattern recognition tolearn from structural object representations such as graphs and strings.These procedures are also applicable to the recognition of histograms,spectra, images and time sequences taking into account the connectivityof samples (bins, wavelengths, pixels or time samples).

The topic of dissimilarity representation is related to the field of non-Mercer kernels in machine learning but it covers a wider set of classifiersand applications. Recently much progress has been made in this area andmany interesting applications have been studied in medical diagnosis,seismic and hyperspectral imaging, chemometrics and computer vision.This review paper offers an introduction to this field and presents anumber of real world applications1.

1 Introduction

In the totality of the world around us we are able to recognize events or objectsas separate items distinguished from their surroundings. We recognize the songof a bird in the noise of the wind, an individual tree in the wood, a cup on thetable, a face in the crowd or a word in the newspaper. Two steps can now bedistinguished. First, the objects are detected in their totality. Second, the isolated

1 We acknowledge financial support from the FET program within the EU FP7, underthe SIMBAD project (contract 213250) as well as the Engineering and PhysicalSciences Research Council in the UK.

C. San Martin and S.-W. Kim (Eds.): CIARP 2011, LNCS 7042, pp. 1–24, 2011.c© Springer-Verlag Berlin Heidelberg 2011

2 R.P.W. Duin and E. P ↪ekalska

object is recognized. These two steps are strongly interconnected and verified byeach other. Only after a satisfactory recognition the detection takes place. Itmay even be questioned whether it is not artificial to make a distinction of theseprocesses in the human recognition of interesting items in the surrounding world.

It is common to separate the two processes in the design of artificial recogni-tion systems. This is possible and fruitful as it is known what type of objects areconsidered in most applications. For example, we know that the system underconstruction has to recognize faces and is not intended to recognize charactersor objects such as cups. The detection step is thereby simplified to selectivelyfocus on faces only, on characters only or on cups only. The recognition step,however, may now lack important information from the context: the recognitionof an isolated character is more difficult than its recognition given the entireword. Recognition systems that take the context into account can become moreaccurate, albeit at the price of a higher complexity.

On the level of the recognition of a single object a similar observation canmade. In the traditional pattern recognition approaches this is mainly done bydescribing objects by isolated features. These are object properties that are ap-propriate locally, at some position on the object, e.g. the sharpness of a corner,or by global properties that describe just a single aspect such as the weight orsize of the object. After these features are determined in a first step, the class orthe name of the object is determined: it is a cup and not an ashtray, or it is thecharacter ’C’ out of the character set in the alphabet. Again it can be doubtedwhether these steps reflect the human recognition process.

Is it really true that we consciously observe a set of features before we cometo a decision? Can we really name well-defined properties that distinguish a cupfrom an ashtray, or John from Peter? Some experts who have thought this overfor their field of expertise may come a long way. Many people, however, canperfectly perform a recognition task, but can hardly name specific features thatserved the purpose. It is only under pressure when they mention some features.

In general, the process of human decision making may not be based on cleararguments but on an unconscious intuition, instead. Arguments or justificationsmay be generated afterwards. They may even be disputed and refuted withoutchanging the decision. This points in the direction that human recognition anddecision making are global processes. These processes take the entire object orsituation into account and a specification into isolated observations and argu-ments becomes difficult.

The above reasoning raises the question whether it is possible to constitutean automatic pattern recognition procedure that is based on the totality of anobject. In this paper some steps in this direction are formulated on the basis ofthe dissimilarity representation. A review will be given of the research that isdone by the authors and their colleagues. Although many papers and experimentswill be mentioned that describe their work, it should be emphasized that thecontext of the research has been of significant importance. The publications andremarks by researchers such as Goldfarb [29], Bunke [44], Hancock and Wilson[40,63], Buhman and Roth [36], Haasdonk [30], Mottle [41], Edelman [25] and

The Dissimilarity Representation for Structural Pattern Recognition 3

Vapnik [58] have been a significant source of inspiration on this topic. It ishowever the aim of this paper to sketch our own line of reasoning in such a waythat it may serve as inspiration for newcomers in this area. We will thereforerestrict this paper to an intuitive explanation illustrated by some examples. Moredetails can be found in the references. Parts of this paper have been extractedfrom a recent journal paper [20] which deals with the same topic but which ismore dedicated to research results and in which less effort has been made tointroduce ideas and concepts carefully.

A global description of objects, which takes their totality into account, shouldbe based on knowledge of how all aspects of the object contribute to the way itappears to the observer. To make this knowledge explicit some structural modelmay be used, e.g. based on graphs. This is not a simple task and usually demandsmuch more background knowledge of the application area than the definition ofsome local properties such as features. The feature-based approach is mainly aneffort in measuring the properties of interest in the observations as presentedby the sensors. As features describe objects just partially, objects belonging todifferent classes may share the same feature vectors. This overlap has to be solvedby a statistical analysis. The two approaches, mentioned above, are linked to thetwo subfields: structural and statistical pattern recognition.

The possibility to merge the two fields has intrigued various researchers overthe decades. Thereby, it has been a research topic from the early days of pat-tern recognition. Originally, most attempts have been made by modifying thestructural approach. Watanabe [59] and especially Fu [26] pointed to severalpossibilities using information theoretic considerations and stochastic syntacti-cal descriptions. In spite of their inspiring research efforts, it hardly resultedin practical applications. Around 1985 Goldfarb [29] proposed to unify the twodirections by replacing the feature-based representation of individual objects bydistances between structural object models. As this proposal hardly requires achange of the existing structural recognition procedures, it may be consideredas an attempt to bridge the gap between the two fields by approaching it fromthe statistical side. Existing statistical tools might thereby become available inthe domain of structural pattern recognition. This idea did not attract muchattention as it was hardly recognized as a profitable approach.

After 1995, the authors of this paper started to study this proposal further.They called it the dissimilarity representation as it allows various non-metric,indefinite or even asymmetric measures. The first experiences were published ina monograph in 2005, [49]. An inspiration for this approach was also the aboveexplained observation that a human observer is primary triggered by object dif-ferences (and later similarities) and that the description by features and modelscomes second; see [25]. The analysis of dissimilarities, mainly for visualization,was already studied in the domain of psychonomy in the 1960s, e.g. by Shepard[54] and Kruskal [35]. The emphasis of the renewed interest in dissimilarities inpattern recognition, however, was in the construction of vector spaces that aresuitable for training classifiers using the extensive toolboxes available in multi-variate statistics, machine learning and pattern recognition. The significance for


the accessibility of these tools in structural object recognition was recognized byBunke [55,44] and others such as Hancock and Wilson [64] and Mottle [42,41].

Before introducing the further contents, let us first summarize key advantagesand drawbacks of using the dissimilarity representation in statistical learning:

– Powerful statistical pattern recognition approaches become available forstructural object descriptions.

– It enables the application expert to use model knowledge in a statisticalsetting.

– As dissimilarities can be computed on top of a feature-based description,the dissimilarity approach may also be used to design classifiers in a featurespace. These classifiers perform very well in comparative studies [23].

– As a result, structural and feature-based information can be combined.– Insufficient performance can often be improved by more observations without

changing the dissimilarity measure.– The computational complexity during execution, i.e. the time spent on the

classification of new objects, is adjustable.– The original representation can be large and computationally complex as

dissimilarities between all object pairs may have to be computed. There areways to reduce this problem [48,39,13].

In this paper we present an intuitive introduction to dissimilarities (Sec. 2), waysto use them for representation (Sec. 3), the computation of classifiers (Sec. 4),the use of multiple dissimilarities (Sec. 5) and some applications (Sec. 6). Thepaper is concluded with a discussion of problems under research.

2 Dissimilarities

Suppose we are given an object to be recognized. That means: can we name it,or can we determine a class of objects of which it belongs to? Some representa-tion is needed if we want to feed it to a computer for an automatic recognition.Recognition is based on a comparison with previous observations of objects likethe one we have now. So, we have to search through some memory. It wouldbe great if an identical object could be found there. Usually, an exact match isimpossible. New objects or their observations are often at least slightly differentfrom the ones previously seen. And this is the challenge of pattern recognition:can we recognize objects that are at most similar to the examples that we havebeen given before? This implies that we need at least the possibility to expressthe similarity between objects in a quantitative way. In addition, it is not alwaysadvantageous to look for an individual match. The generalization of classes ofobjects to a ’concept’, or a distinction which can be expressed in a simple classi-fication rule is often faster, demands less memory and/or can be more accurate.

It has been observed before [25], and it is in line with the above discussions,that in human recognition processes it is more natural to rely on similaritiesor dissimilarities between objects than to find explicit features of the objectsthat are used in the recognition. This points to a representation of objects based


on a pairwise comparison of the new examples with examples that are alreadycollected. This differs from the feature-based representations that constitute thebasis of the traditional approaches to pattern recognition described in the well-known textbooks by Fukunaga [27], Duda, Hart and Stork [17], Devijver andKittler [16], Ripley [53], Bishop [7],Webb [60] and Theodorides [56]. We wantto point out that although the pairwise dissimilarity representation presentedhere is different in its foundation from the feature-based representation, manyprocedures described in the textbooks can be applied in a fruitful way.

We will now assume that a human recognizer, preferably an expert w.r.t. theobjects of interest, is able to formulate a dissimilarity measure between objectsthat reflects his own perception of object differences (for now we will stick todissimilarity measures). A dissimilarity measure d(oi, oj) between two objects oi

and oj out of a training set of n objects may have one or more of the followingproperties for all i, j, k ≤ n.

– Non-negativity: d(oi, oj) ≥ 0.– Identity of indiscernibles: d(oi, oj) = 0 if and only if oi ≡ oj .– Symmetry: d(oi, oj) = d(oj , oi).– Triangle inequality: d(oi, oj) + d(oj , ok) ≥ d(oi, ok).– Euclidean: An n × n dissimilarity matrix D is Euclidean if there exists

an isometric Euclidean embedding into a Euclidean space. In other words, aEuclidean space with n vectors can be found such that the pairwise Euclideandistances between these vectors are equal to the original distances in D.

– Compactness: A dissimilarity measure is defined here as compact if a suf-ficiently small perturbation of an object (from a set of allowed transforma-tions) leads to an arbitrary small dissimilarity value between the disturbedobject and its original. We call such a measure compact because it results incompact class descriptions for which sufficiently small disturbances will notchange the class membership of objects. Note that this definition is differentthan compactness discussed in topological spaces.

The first two properties together produce positive definite dissimilarity measures.The first four properties coincide with the mathematical definition of a metricdistance measure.

Non-negativity and symmetry seem to be obvious properties, but sometimesdissimilarity measures are defined otherwise. E.g. if we define the distance to acity as the distance to the border of that city, then a car that reaches the borderfrom outside has a distance zero. When the car drives further into the city thedistance may be counted as negative in order to keep consistency. An exampleof an asymmetric distance measure is the directed Hausdorff distance betweentwo sets of points A and B: dH(A, B) = maxx{miny{d(x, y), x ∈ A, y ∈ B}}.

An important consequence of using positive definite dissimilarity measures isthat classes are separable for unambiguously labeled objects (identical objectsbelong to the same class). This directly follows from the fact that if two objectshave a distance zero they should be identical and as a consequence they belongto the same class. For such classes a zero-error classifier exists (but may still be


difficult to find). See [49]. This is only true if the dissimilarity measure reflects allobject differences. Dissimilarity measures based on graphs, histograms, featuresor other derived measurements may not be positive definite as different objectsmay still be described by the same graph (or histogram, or sequence) and therebyhave a zero dissimilarity.

The main property is the Euclidean property. A metric distance measure infact states that the Euclidean property holds for every set of three points whilethe first two properties (positive definiteness) state that the dissimilarity of everypair of points is Euclidean.

We may distinguish the properties of the dissimilarity measure itself and theway it works out for a set of objects. The first should be analyzed mathematicallyfrom the definition of the measure and the known properties of the objects. Thesecond can be checked numerically from a given n × n dissimilarity matrix D.There might be a discrepancy between what is observed in a finite data matrixand the definition of the measure. It may occur for instance that the matrix D fora given training set of objects is perfectly Euclidean but that the dissimilaritiesfor new objects behave differently.

The concept of compactness is important for pattern recognition. It was firstused in the Russian literature around 1965, e.g. see Arkedev and Braverman [3],and also [18]. We define here that a compact class of objects consists of a finitenumber of subsets, such that in each subset every object can be continuouslytransformed (morphed) into every other object of that subset without passingthrough objects outside the subset. This property of compactness is slightly dif-ferent from the original concept defined in [3] where it is used as a hypothesis onwhich classifiers are defined. It is related to the compactness used in topology.Compactness is a basis for generalization from examples. Without proof we statehere that for compact classes the consequence of the no-free-lunch theorem [65](every classifier is as good as any other classifier unless we use additional knowl-edge) is avoided: compactness pays the lunch. The prospect is that for the caseof positive definite dissimilarity measures and unambiguously labeled objects,the classes can be separated perfectly by classifiers of a finite complexity.

3 Representation

Arepresentation of real world objects is needed in order to be able to relate to them.It prepares the generalization step by which new, unseen objects are classified. So,the better the representation, the more accurate classifiers can be trained. The tra-ditional representation is defined by numerical features. The use of dissimilaritiesis an attractive alternative, for which arguments were given in Introduction. Thissection provides more details by focussing on the object structure.

3.1 Structural Representations

The concept of structure is ill defined. It is related to the global connectivity ofall parts of the object. An image can be described by a set of pixels organized in


a square grid. This grid may be considered as the structure of the image. It is,however, independent of the content of the image. If this is taken into accountthen the connectivity between the pixels may be captured by weights, e.g. relatedto the intensity values of the neighboring pixels. We may also forget that thereare pixels and determine regions in the image by a segmentation procedure. Thestructure may then be represented by a graph in which every node is relatedto an image segment and the graph edges correspond to neighboring segments.Nodes and edges may have attributes that describe properties of the segmentsand the borders between them.

A simpler example of a structure is the contour of an image segment or ablob: its shape. The concept of shape leads to a structure, but shapes are alsocharacterized by features, e.g. the number of extremes or a set of moments. Astructural representation of a shape is a string. This is a sequence of symbolsrepresenting small shape elements, such as straight lines (in some direction) orcurves with predefined curvatures. Shapes are also found in spectra, histogramsand time signals. The movement of an object or a human body may be describedas a set of coordinates in a high-dimensional space as a function of time. Thismulti-dimensional trajectory has a shape and may be considered as a structure.

The above examples indicate that structures also have some (local) propertiesthat are needed for their characterization. Examples of pure structures withoutattributes can hardly be found. Certainly, if we want to represent them in a waythat facilitates comparisons, we will use attributes and relations (connections).The structural representations used here will be restricted to attributed graphsand sequences.

3.2 The Dissimilarity Representation

Dissimilarities themselves have been discussed in Sec. 2. Three sets of objectsmay be distinguished for constructing a representation:

– A representation set R = {r1, . . . , rk}. These are the objects we refer to.The dissimilarities to the representation set have to be computed for trainingobjects as well as for test objects used for evaluation, or any objects to beclassified later. Sometimes the objects in the set R are called prototypes.This word may suggest that these objects are in some way typical examplesof the classes. That can be the case but it is not necessary. So prototypesmay be used for representation, but the representation set may also consistof other objects.

– A training set T = {o1, . . . , on}. These are the objects that are used totrain classifiers. In many applications we use T := R, but R may also be justa (small) subset of T , or be entirely different from T .

– A test set S. These are the objects that are used to evaluate classificationprocedure. They should be representative for the target objects for whichthe classification procedure is built.

After determining these three sets of objects the dissimilarity matrices D(T, R)and D(S, R) have to be computed. Sometimes also D(T, T ) is needed, e.g. when


the representation set R ⊂ T has to be determined by a specific algorithm. Thenext problem is how to use these two or three matrices for training and testing.Three procedures are usually considered:

– The k-nearest neighbor classifier. This is the traditional way to classifynew objects in the field of structural pattern recognition: assign new objectsto the (majority) class of its (k) nearest neighbor(s). This procedure candirectly by applied to D(S, T ). The dissimilarities inside the training set,D(T, T ) or D(T, R) are not used.

– Embedded space. Here a vector space and a metric (usually Euclidean) aredetermined from D(T, T ) containing n = |T | vectors, such that the distancesbetween these vectors are equal to the given dissimilarities. See Sec. 3.3 formore details.

– The dissimilarity space. This space is postulated as a Euclidean vectorspace defined by the dissimilarity vectors d(·, R) = [d(·, r1), . . . , d(·, rk)]T

computed to the representation set R as dimensions. Effectively, the dissim-ilarity vectors are used as numerical features. See Sec. 3.4.

3.3 Embedding of Dissimilarities

The topic of embedding dissimilarity matrices has been studied for a long time.As mentioned in the introduction (Sec. 1), it was originally used for visualiz-ing the results of psychonomic experiments and other experiments representingdata in pairwise object comparisons [54,35]. In such visualization tasks, a reli-able, usually 2D map of the data structure is of primary importance. Variousnonlinear procedures have been developed over the years under the name ofmulti-dimensional scaling (MDS) [9].

It is difficult to reliably project new data to an existing embedded spaceresulting from a nonlinear embedding. Therefore, such embeddings are unsuitablefor pattern recognition purposes in which a classifier trained in the embeddedspace needs to be applied to new objects. A second, more important drawbackof the use of nonlinear MDS for embedding is that the resulting space does notreflect the original distances anymore. It usually focusses either on local or globalobject relations to force a 2D (or other low-dimensional) result.

For the purpose of generalization a restriction to low-dimensional spaces isnot needed. Moreover, for the purpose of the projection of new objects linearprocedures are preferred. Therefore, the linear MDS embedding has been studied,also known as classical scaling [9]. As the resulting Euclidean space is by itsvery nature not able to perfectly represent non-Euclidean dissimilarity data, seeSec. 2, a compromise has to be made. The linear Euclidean embedding procedureis based on an eigenvalue decomposition of the Gram matrix derived from thegiven n×n dissimilarity matrix D, see [29,49], in which some eigenvalues becomenegative for non-Euclidean dissimilarities. This conflicts with the construction ofa Euclidean space as these eigenvalues are related to variances of the extractedfeatures, which should be positive. This is solved in classical scaling by neglectingall ’negative’ directions. The distances in this embedded space may thereby beentirely different from the original dissimilarity matrix D.


The approach followed by the pseudo-Euclidean embedding is to constructa vector space [49] in which the metric is adjusted such that the squared dis-tance contributions of the ’negative’ eigenvectors are counted as negative. Theresulting pseudo-Euclidean space thereby consists out of two orthogonal Eu-clidean spaces of which the distances are not added (in the squared sense) butsubtracted. Distances computed in this way are exactly equal to the original dis-similarities, provided that they are symmetric and self-dissimilarity is zero. Suchan embedding is therefor an isometric mapping of the original D into a suitablepseudo-Euclidean space, which is an inner product space with an indefinite innerproduct.

The perfect representation of D in a pseudo-Euclidean embedded space is aninteresting proposal, but it is not free from some disadvantages:

– Embedding relies on a square dissimilarity matrix, usually D(T, T ). The dis-similarities between all pairs of training objects should be taken into account.The computation of this matrix as well as the embedding procedure itselfmay thereby be time and memory demanding operations.

– Classifiers that obey the specific metric of the Pseudo-Euclidean space are dif-ficult to construct or not yet well defined. Some have been studied [32,50,21],but many problems remain. For instance, it is not clear how to define a normaldistribution in a pseudo-Euclidean space. Also the computation of SVM maybe in trouble as the related kernel is indefinite, in general [31]. Solutions areavailable for specific cases. See also Sec. 4.

– There is a difficulty in a meaningful projection of new objects to an existingpseudo-Euclidean embedded space. The straightforward projection opera-tions are simple and linear, but they may yield solutions with negative dis-tances to other objects even though the original distances are non-negative.This usually happens when a test object is either an outlier or not well rep-resented in the training set T (which served to define the embedded space).A possible solution is to include such objects in the embedding procedureand retrain the classifier for the new objects. For test objects this impliesthat they will participate in the representation. Classification may therebyimprove at the cost of the retraining. This approach is also known as trans-ductive learning [58].

– The fact that embedding strictly obeys the given dissimilarities is not al-ways an advantage. All types of noise and approximations related to thecomputation of dissimilarities are expressed in the result. It may thereby bequestioned whether all non-Euclidean aspects of the data are informative. In[19] it is shown that there are problems for which this is really the case.

In order to define a proper topology and metric, mathematical texts, ↪e.g. [8],propose to work with the associated Euclidean space instead of the pseudo-Euclidean space. In this approach all ’negative’ directions are treated as ’positive’ones. As a result, one can freely use all traditional classifiers in such a space.The information extracted from the dissimilarity matrix is used but the originaldistance information is not preserved and may even be significantly distorted.Whether this is beneficial for statistical learning depends on the problem.


3.4 The Dissimilarity Space

The dissimilarity space [46,49] postulates a Euclidean vector space defined bythe dissimilarity vectors. The elements of these vectors are dissimilarities froma given object to the objects in the representation set R. The dissimilarity vec-tors serve as features for the objects in the training set. Consequently, such aspace overcomes all problems that usually arise with the non-Euclidean dissim-ilarity measures, simply by neglecting the character of the dissimilarity. Thisapproach is at least locally consistent for metric distance measures: distancesin the dissimilarity space between pairs of objects characterized by small dis-similarities d(oi, oj) will also have a small distance as their dissimilarity vectorsd(oi, R) = [d(oi, r1), . . . , d(oi, rk)]T and d(oj , R) = [d(oj , r1), . . . , d(oj , rk)]T willbe about equal. This may serve as a proof that the topology of a set of objectswith given dissimilarities {d(oi, oj)}i,j=1:n is identical to the topology of this setof objects in the dissimilarity space {dE(d(oi, R), d(oj , R))}i,j=1:nprovided thatR is sufficiently large (to avoid that different objects have, by accident, a zerodistance in the dissimilarity space).

If all training objects are used for representation, the dimension of the dis-similarity space is equal to |T |. Although, in principle, any classifier defined fora feature space may be applied to the dissimilarity space, some of them willbe ill-defined or overtrained for such a large representation set. Dimension re-duction, e.g. by prototype selection may thereby be an important issue in thisapproach [48,39,13]. Fortunately, these studies show that if the reduction is notput to the extreme, a randomly selected representation set may do well. Herethe dissimilarity space is essentially different from a traditional feature space:features may be entirely different in their nature. A random selection of R mayexclude a few significant examples. The objects in a training set, however, willin expectation include many similar ones. So, a random selection is expected tosample all possible aspects of the training set, provided that the training set Tas well as the selected R are sufficiently large.

If a representation set R is a subset of T and we use the complete set T intraining, the resulting representation D(T, R) contains some zero dissimilaritiesto objects in R. This is not expected to be the case for new test objects. Inthat sense the training objects that participate in the representation set are notrepresentative for test objects. It might be better to exclude them. In all ourexperiments however we found just minor differences in the results if we usedD(T \R, R) instead of D(T, R).

Although any feature-based classifier can be used in a dissimilarity space,some fit more naturally than others. For that reason we report a number ofexperiments and their findings in Sec. 4.

4 Classifiers

We will discuss here a few well-known classifiers and their behavior in variousspaces. This is a summary of our experiences in many studies and applications.See [49] and its references.


In making a choice between embedding and the dissimilarity space for train-ing a classifier one should take into account the essential differences betweenthese spaces. As already stated, embedding strictly obeys the distance charac-teristics of the given dissimilarities, while the dissimilarity spaces neglects this.In addition, there is a nonlinear transformation between these spaces: by com-puting the distances to the representation objects in the embedded space thedissimilarity space can be defined. As a consequence, a linear classifier in theembedded space is a nonlinear classifier in the dissimilarity space, and the otherway around. Comparing linear classifiers computed in these spaces is therebycomparing linear and nonlinear classifiers.

It is outside the scope of this paper, but the following observation might behelpful for some readers. If the dissimilarities are not constructed by a procedureon a structural representation of objects, but are derived as Euclidean distancesin a feature space, then the pseudo-Euclidean embedding effectively reconstructsthe original Euclidean feature space (except for orthonormal transformations).So in that case a linear classifier in the dissimilarity space is a nonlinear classifierin the embedded space, which is the same nonlinear classifier in the feature space.Such a classifier, computed in a dissimilarity space, can perform very well [23].

4.1 Nearest Neighbor Classifier

The k-nearest neighbor (k-NN) classifier in an embedded (pseudo-)Euclideanspace is based on the distances computed in this space. By definition these arethe original dissimilarities (provided that the test examples are embedded to-gether with the training objects). So without the process of embedding this clas-sifier, can directly be applied to a given dissimilarity matrix. This is the classifiertraditionally used by many researchers in the area of structural pattern recog-nition. The study of the dissimilarity representation arose because this classifierdoes not make use of the given dissimilarities in the training set. Classification isentirely based on the dissimilarities of a test object to the objects in the training(or representation) set only.

The k-NN rule computed in the dissimilarity space relies on a Euclidean dis-tance between the dissimilarity vectors, hence the nearest neighbors are deter-mined by using all dissimilarities of a given object to the representation objects.As explained in Sec. 3.4 for the metric case and for large sets it is expectedthat the distances between similar objects are small for the two spaces. So, it isexpected that learning curves are asymptotically identical, but for small trainingsets the dissimilarity space works better as it uses more information.

4.2 Parzen Density Classifiers

The class densities computed by the Parzen kernel density procedure are basedon pairwise distance computations between objects. The applicability of thisclassifier as well as its performance is thereby related to those of the k-NN rule.Differences are that this classifier is more smooth, depending on the choice ofthe smoothing parameter (kernel) and that its optimization involves the entiretraining set.


4.3 Normal Density Bayes Classifiers

Bayes classifiers assume that classes can be described by probability densityfunctions. Using class priors and Bayes’ rule the expected classification error isminimized. In case of normal density function either a linear classifier (LinearDiscriminant Analysis, LDA) arises on the basis of equal class covariances, ora quadratic classifier is obtained for the general case (Quadratic DiscriminantAnalysis, QDA). These two classifiers belong to best possible in case of (closeto) normal class distributions and a sufficiently large training set. As mean vec-tors and covariance matrices can be computed in a pseudo-Euclidean space, see[29,49], these classifiers exist there as well if we forget the starting point of nor-mal distributions. The reason is that normal distributions are not well definedin pseudo-Euclidean spaces; it is not clear what a normal distribution is unlesswe refer to associated Euclidean spaces.

In a dissimilarity space the assumption of normal distributions works oftenvery well. This is due to the fact that many cases dissimilarity measures arebased on, or related to sums of numerical differences. Under certain conditionslarge sums of random variables tend to be normally distributed. It is not per-fectly true for distances as we often get Weibull [12] or χ2 distributions, but theapproximations are sufficient for a good performance of LDA and QDA. Theeffect is emphasized if the classification procedure involves the computation oflinear subspaces, e.g. by PCA. Thanks to projections normality is emphasizedeven more.

4.4 Fisher’s Linear Discriminant

In a Euclidean space the Fisher linear discriminant (FLD) is defined as the linearclassifier that maximizes the Fisher criterion, i.e. the ratio of the between-classvariance to the within-class variance. For a two-class problem, the solution isequivalent to LDA (up to an added constant), even though no assumption ismade about normal distributions. Since variance and covariance matrices arewell defined in pseudo-Euclidean spaces, the Fisher criterion can be used toderive the FLD classifier there. Interestingly, FLD in a pseudo-Euclidean spacecoincides with FLD in the associated Euclidean space. FLD is a linear classifierin a pseudo-Euclidean space, but can be rewritten to FLD in the associatedspace; see also [50,32].

In a dissimilarity space, which is Euclidean by definition, FLD coincides withLDA for a two-class problem. The performance of these classifiers may differfor multi-class problems as the implementations of FLD and LDA will usuallyvary then. Nevertheless, FLD performs very well. Due to the nonlinearity of thedissimilarity measure, FLD in a dissimilarity space corresponds to a nonlinearclassifier in the embedded pseudo-Euclidean space.

4.5 Logistic Classifier

The logistic classifier is based on a model of the class posterior probabilities as afunction of the distance to the classifier [1]. The distance between a vector and a


linear hyperplane in a pseudo-Euclidean space however is an unsuitable conceptfor classification as it can have any value (−∞,∞) for vectors on the same side ofthis hyperplane. We are not aware of a definition and an implementation of thelogistic classifier for pseudo-Euclidean spaces. Alternatively, the logistic classifiercan be constructed in the associated Euclidean space.

In a dissimilarity space, the logistic classifier performs well, although normaldensity based classifiers work often better. It relaxes the demands for normalityas made by LDA. It is also more robust in case of high-dimensional spaces.

4.6 Support Vector Machine (SVM)

The linear kernel in a pseudo-Euclidean space is indefinite (non-Mercer). Thequadratic optimization procedure used to optimize a linear SVM may therebyfail [30]. SVM can however be constructed if the contribution of the positivesubspace of the Euclidean space is much stronger than that of the negativesubspace. Mathematically, it means that the measure is slightly deviating fromthe Euclidean behavior and the solution of SVM optimization is found in thepositive definite neighborhood. Various researchers have reported good results inapplying this classifier, e.g. see [11]. Although the solution is not guaranteed andthe algorithm (in this case LIBSVM, [14]) does not stop in a global optimum, agood classifier can be obtained.

In case of a dissimilarity space the (linear) SVM is particularly useful for com-puting classifiers in the complete space for which R := T . The given training setdefines therefore a separable problem. The SVM does not or just hardly over-train in this case. The advantage of this procedure is that it does not demand areduction of the representation set. A linear SVM is well defined. By normaliz-ing the dissimilarity matrix (such that the average dissimilarity is one) we foundstable and good results in many applications by setting the trade-off parameterC in the SVM procedure [15] to C = 100. Hereby, additional cross-validationloops are avoided to optimize this parameter. As a result, in an application onecan focus on optimizing the dissimilarity measure.

4.7 Combining Classifiers

In the area of dissimilarity representations many approaches can be considered.Various strategies can be applied for the choice of the representation set, eitherembedded or dissimilarity spaces can be used, and various modifications can beconsidered, e.g. refinements or correction procedures for these spaces; see [24,21].Instead of selecting one of the approaches, classifier combining may provide anadditional value. However, as all these classifiers are based on the same dissimi-larities they do not provide any additional or valuable information. Effectively,just additional procedures are considered that encode different nonlinearities. Asthe given square dissimilarity matrix D describes an already linearly separableset of objects (under the assumption of the positive definite dissimilarity) wedo not expect that in general much can be gained by combining, although anoccasional success is possible in particular problems.


5 Multiple Dissimilarities

Instead of generating sets of classifiers defined on the same dissimilarity rep-resentation also modifications of the dissimilarity measure may be considered.Another measure can emphasize other aspects of the objects. The resulting dis-similarity matrices cannot be derived from each other, in general. Consequently,they are chosen to encode different information. Combining various dissimilarityrepresentations or classifiers derived from them is now much more of interest.These types of studies are closely related to the studies on kernel metric learning[61,68,66]. An important difference is that the study of kernels is often focussedon the use of SVM for classification, and consequently positive definite kernelsobeying the Mercer conditions are the key. As the dissimilarity representationpermits many classifiers this point is not relevant for dissimilarity measures.On the contrary, the unrestricted use of dissimilarity definitions is of particularsignificance for structural pattern recognition as there non-Euclidean measuresnaturally arise. See also [22].

There are a number of reasons why a set of different dissimilarities betweenobjects arises. A few examples are:

– The same set of objects is observed multiple times under different conditions.– The dissimilarities are computed on different samplings from the original

signals (multi-scale approach).– Different dissimilarity measures are used on the same signals.

A very interesting observation that can be made from various studies such as[33,57] is that a simple element-wise averaging of dissimilarity matrices definedby different measures often leads to a significant improvement of the classificationerror over the best individual measure. Attempts to improve this further by aweighted averaging is sometimes successful but often appears not to be useful.The precise value of the weights does not seem to be very significant, either.

6 Application Examples

In this section we will discuss a few examples that are typical for the possibilitiesof the use of dissimilarities in structural pattern recognition problems. Some havebeen published by us before [22] for another readership. They are repeated hereas they may serve well as an illustration in this paper.

6.1 Shapes

A simple and clear example of a structural pattern recognition problem is therecognition of blobs: 2D binary structures. An example is given in Fig. 1. It isan object out of the five-class chickenpieces dataset consisting of 445 images [2].One of the best structural recognition procedure uses a string representationof the contour described by a set of segments of the same length [10]. Thestring elements are the consecutive angles of these segments. The weighted edit


5 10 15 20 25 30 35 400

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Fig. 1. Left: some examples of the chickenpieces dataset. Right: the error curves as afunction of the segment length L.

distances between all pairs of contours are used to compute dissimilarities. Thismeasure is non-Euclidean. A (γ, �L) family of problems is considered dependingon the specific choice for the cost of one editing operation γ as well as for thesegment’s length L used in the contour description. As a result, the classificationperformance depends on the parameters used, as shown in Fig 1, right. 10-foldcross-validation errors are shown there for the 1-NN rule directly applied on thedissimilarities as well as the results for the linear SVM computed by LIBSVM,[14], in the dissimilarity space. In addition the results are shown for the averageof the 11 dissimilarity matrices. It is clearly observed that the linear classifier inthe dissimilarity space (SVM-1) improves the traditional 1-NN results and thatcombining the dissimilarities improves the results further on.

6.2 Histograms and Spectra

Histograms and spectra offer very simple examples of data representations thatare judged by human experts on their shape. In addition, also the sampling of thebins or wavelengths may serve as a useful vector representation for an automaticanalysis. This is thanks to the fact that the domain is bounded and that spectraare often aligned. Below we give an example in which the dissimilarity represen-tation outperforms the straightforward vector representation based on samplingbecause the first can correct for a wrong calibration (resulting in an imperfectalignment) in a pairwise fashion. Another reason to prefer dissimilarities for his-tograms and spectra over sampled vectorial data is that a dissimilarity measureencodes shape information. For examples see the papers by Porro [52,51].

We will consider now a dataset of 612 FL3-A DNA flow cytometer histogramsfrom breast cancer tissues in a resolution of 256 bins. The initial data wereacquired by M. Nap and N. van Rodijnen of the Atrium Medical Center inHeerlen, The Netherlands, during 2000-2004, using the four tubes 3-6 of a DACOGalaxy flow cytometer. Histograms are labeled into three classes: aneuploid (335patients), diploid (131) and tetraploid (146). We averaged the histograms of thefour tubes thereby covering the DNA contents of about 80000 cells per patient.


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HistogramsEuclidean DistancesCalibrated Distances

Fig. 2. Examples of some flow cytometer histograms: aneuploid, diploid and tetraploid.Bottom right shows the learning curves.

We removed the first and the last bin of every histogram as here outliers arecollected, thereby obtaining 254 bins per histogram. Examples of histograms areshown in Fig. 2. The following representations are used:

Histograms. Objects (patients) are represented by the normalized values ofthe histograms (summed to one) described by a 254-dimensional vector. Thisrepresentation is similar to the pixel representation used for images as it isbased on just a sampling of the measurements.

Euclidean distances. These dissimilarities are computed as the Euclideandistances in the vector space mentioned above. Every object is representedby by a vector of distances to the objects in the training set.

Calibrated distances. As the histograms may suffer from an incorrect cali-bration in the horizontal direction (DNA content) for every pairwise dissim-ilarity we compute the multiplicative correction factor for the bin positionsthat minimizes their dissimilarity. Here we used the �1 distance. This repre-sentation makes use of the shape structure of the histograms and removesan invariant (the wrong original calibration).

A linear SVM with a fixed trade-off parameter C is used in learning. The learningcurves for the three representations are shown in the bottom right of Fig. 2. Theyclearly illustrate how for this classifier the dissimilarity representation leads to


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Fig. 3. Left: examples of the images used for the digit recognition experiment. Right:the learning curves.

better results than the vector representation based on the histogram sampling.The use of the background knowledge in the definition of the dissimilarity mea-sure improves the results further.

6.3 Images

The recognition of objects on the basis of the entire image can only be doneif these images are aligned. Otherwise, earlier pre-procession or segmentation isnecessary. This problem is thereby a 2-dimensional extension of the histogramand spectra recognition task. We will show an example of digit recognition byusing a part of the classic NIST database of handwritten numbers [62] on thebasis of random subsets of 500 digits for the ten classes 0-9. The images wereresampled to 32 × 32 pixels in such a way that the digits fit either horizontallyor vertically. Fig. 3 shows a few examples: black is ’1’ and white is ’0’. Thedataset is repeatedly split into training and test sets and hold-out classificationis applied. In every split the ten classes are evenly represented.The following representations are used:

Features. We used 10 moments: the seven rotations invariant moments andthe moments [00], [01], [10], measuring the total number of black pixels andthe centers of gravity in the horizontal and vertical directions.

Pixels. Every digit is represented by a vector of the intensity values in 32∗32 =1024 dimensional vector space.

Dissimilarities to the training object. Every object is represented by theEuclidean distances to all objects in the training set.

Dissimilarities to blurred digits in the training set. As the pixels in thedigit images are spatially connected blurring may emphasize this. In thisway the distances between slightly rotated, shifted or locally transformedbut otherwise identical digits becomes small.

The results are shown in Fig. 3 on the right. They show that the pixel rep-resentation is superior for large training sets. This is to be expected as thisrepresentation stores asymptotically the universe of possible digits. For small


training sets a suitable set of features may perform better. The moments we usehere are very general features. Better ones can be found for digit description.As explained before a feature-based description reduces the (information on the)object: it may be insensitive for some object modifications. For sufficiently largerepresentation sets the dissimilarity representation may see all object differencesand may thereby perform better.

6.4 Sequences

The recognition of sequences of observations is in particular difficult if the se-quences of a given class vary in length, but capture the same ’story’ (information)from the beginning to the end. Some may run faster, or even run faster over justa part of the story and slow down elsewhere. A possible solution is to rely onDynamic Time Warping (DTW) that relates the sequences in a nonlinear way,yet obeys the order of the events. Once two sequences are optimally aligned, thedistance between them may be computed.

An example in which the above has been applied successfully is the recognitionof 3-dimensional gestures from the sign language [38] based on an statisticallyoptimized DTW procedure [4]. We took a part of a dataset of this study: the 20classes (signs) that were most frequently available. Each of these classes has 75examples. The entire dataset thereby consists of a 1500× 1500 matrix of DTW-based dissimilarities. The leave-one-out 1-NN error for this dataset is 0.041,which is based on the computation of 1499 DTW dissimilarities per test object.In Fig. 4, left, a scatterplot is shown of the first two PCA components showingthat some classes can already be distinguished with these two features (linearcombinations of dissimilarities).

We studied dissimilarity representations consisting of just one randomly drawnexampleper class.The resultingdissimilarity spacehas thereby20dimensions.Newobjects have to be compared with just these 20 objects. This space is now filledwith randomly selected training sets of containing between 2 and 50 objects perclass. Remaining objects are used for testing. Two classifiers are studied, the linearSVM (using the LIBSVM package [14]) with a fixed trade-off parameter C = 100(we used normalized dissimilarity matrices with average dissimilarities of 100) andLDA. The experiment was repeated 25 times and the results averaged out. Thelearning curves in Fig. 4, right, show the constant value of the 1-NN classifier per-formance using the dissimilarities to the single training examples per class only, andthe increasing performances of the two classifiers for a growing number of trainingobjects. Their average errors for 50 training objects per class is 0.07. Recall thatthis is still based on the computation of just 20 DTW dissimilarities per object aswe work in the related 20-dimensional dissimilarity space. Our experiments showthatLDAreaches an error of 0.035 for a representation set of three objects per class,i.e. 60 objects in total. Again, the training set size is 50 examples per class, i.e. 1000examples in total. For testing new objects one needs to compute a weighted sum(linear combination) of 60 dissimilarity values giving the error of 0.035 instead ofcomputing and ordering 1500 dissimilarities to all training objects for the 1-NNclassifier leading to an error of 0.041.


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Fig. 4. PCA and learning curves for the 20-class Delft Gesture Dataset

6.5 Graphs

Graphs2 are the main representation for describing structure in observed objects.In order to classify new objects, the pairwise differences between graphs have tobe computed by using a graph matching technique. The resulting dissimilaritiesare usually related to the cost of matching and may be used to define a dis-similarity representation. We present here classification results obtained with asimple set of graphs describing four objects in the Coil database [43] described by72 images for every object. The graphs are the Delaunay triangulations derivedfrom corner points found in these images; see [67]. They are unattributed. Hence,the graphs describe the structure only. We used three dissimilarity measures:

CoilDelftSame. Dissimilarities are found in a 5D space of eigenvectors derivedfrom the two graphs by the JoEig approach; see [37]

CoilDelftDiff. Graphs are compared in the eigenspace with a dimensionalitydetermined by the smallest graph in every pairwise comparison by the JoEigapproach; see [37]

CoilYork. Dissimilarities are found by graph matching, using the algorithm ofGold and Ranguranjan; [28]

All dissimilarity matrices are normalized such that the average dissimilarity is1. In addition to the three dissimilarity datasets we used also their averageddissimilarity matrix.

In a 10-fold cross-validation experiment, with R := T , we use four classifiers:the 1-NN rule on the given dissimilarities and the 1-NN rule in the dissimilarityspace (listed as 1-NND in Table 6.5), LDA on a PCA-derived subspace covering99% of the variance and the linear SVM with a fixed trade-off parameter C = 1.All experiments are repeated 25 times. Table 6.5 reports the mean classificationerrors and the standard deviations of these means in between brackets. Someinteresting observations are:2 Results presented in this section are based on joint research with Prof. Richard

Wilson, University of York, UK, and Dr. Wan-Jui Lee, Delft University of Technol-ogy, The Netherlands.


Table 1. 10-fold cross-validation errors averaged over 25 repetitions

dataset 1-NN 1-NND PCA-LDA SVM-1

CoilDelftDiff 0.477 (0.002) 0.441 (0.003) 0.403 (0.003) 0.395 (0.003)CoilDelftSame 0.646 (0.002) 0.406 (0.003) 0.423 (0.003) 0.387 (0.003)CoilYork 0.252 (0.003) 0.368 (0.004) 0.310 (0.004) 0.326 (0.003)Averaged 0.373 (0.002) 0.217 (0.003) 0.264 (0.003) 0.238 (0.002)

– The CoilYork dissimilarity measure is apparently much better than the twoCoilDelft measures.

– The classifiers in the dissimilarity space however are not useful for theCoilYork measure, but they are for the CoilDelft measures. Apparently thesetwo ways of computing dissimilarities are essentially different.

– Averaging all three measures significantly improves the classifier performancein the resulting dissimilarity space, even outperforming the original bestCoilYork result. It is striking that this does not hold for the 1-NN ruleapplied to the original dissimilarities.

7 Discussion

In this paper we have given a review of the arguments why the dissimilarityrepresentation is useful for applications in structural pattern recognition. Thishas been illustrated by a set of examples on real world data. This all shows thatusing the collective information from all other objects and relating them to eachother on the top of the given pairwise dissimilarities (either in the dissimilarityor embedded space), reveals an additional source of information that is otherwiseunexplored.

The dissimilarity representation makes the statistical pattern recognition toolsavailable for structural data. In addition, features are given the use of combinersmay be considered or the features may be included in the dissimilarity measure.If either the chosen or optimized dissimilarity measure covers all relevant aspectsof the data, then a zero dissimilarity arises if and only if the objects are identical.In that case the classes are separable in a sufficiently large dissimilarity space.Traditional statistical classification tools are designed for overlapping classes.They may still be applied, but the topic of designing proper generalization toolsmay be reconsidered for the case of high-dimensional separable classes. For in-stance, the demand that a training set should be representative for the futuredata to be classified in the statistical sense (i.e. they are generated from the samedistributions) is not necessary anymore. These sets should just cover the samedomain.

A result, not emphasized in this paper, is that for positive definite dissim-ilarity measures, see Sec. 2, and sufficiently complex classifiers, any measureasymptotically (for increasing training and representation sets) reaches a zero-error classifier. So, a poorly discriminative dissimilarity measure can be com-pensated by a large training set as long as the measure is positive definite.


An interesting experimental observation is that if several of these measures aregiven the average of the dissimilarity matrix offers a better representation thanany of them separable. Apparently, the asymptotic convergence speeds (almost)always contribute in combinations and do not disturb each other.

One may wonder whether the dissimilarity measures used in Sec. 6 are allhave the positive definite property. However, entirely different objects may bedescribed by identical histograms or graphs. So, the users should analyze, if theyneed this property and whether an expert is able to label the objects unambigu-ously on the basis of histograms or graphs only. If not, as a way to attain a bettergeneralization, he may try to extend the distance measure with some features,or simply add another, possibly bad measure, which is positive definite.

They area of dissimilarity representations is conceptually closely related tokernel design and kernel classifiers. It is, however, more general as it allowsfor indefinite measures and makes no restrictions w.r.t. the classifier [47,50]. Thedissimilarity representation is essentially different from kernel design in the sensethat the dissimilarity matrix is not necessarily square. This has not only strongcomputational advantages, but also paves the way to the use of various classifiers.As pointed out in Sec. 3.4, systematic prototype selection is mainly relevant toobtain low-dimensional dissimilarity spaces defined by a small set of prototypes.Another way to reach this goal, not discussed here due to space limit, is the useof out-of-the training set prototypes or the so-called generalized dissimilarityrepresentation. Here prototypes are replaced by sets of prototypes, by modelsbased on such sets, or by artificially constructed prototypes; see [5,6,45,34].

For future research in this field we recommend the study of dissimilarity mea-sures for sets of applications such as spectra, images, etcetera. In every individualapplication measures may be optimized for the specific usage, but the availabilityof sets of measures for a broader field of structural applications may, accordingto our intuition, be most profitable for the field of structural pattern recognition.

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